The quantification of the spreading of the hypergeometric orthogonal polynomials (HOPs) along the support interval \(\Lambda \subseteq {\mathbb {R}}\)  is interesting per se in the theory of special functions and approximation theory, and because of their numerous applications in quantum mechanics and mathematical physics [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. A relevant reason for the latter is that the HOPs control the physical solutions of the non-relativistic and relativistic wave equations of a great deal of relevant quantum systems (oscillator-like systems, hydrogenic atoms,...) [1]}, [8]}, [9]}. Here we study the spreading measures of the real HOPs, \(\lbrace p_n(x)\rbrace \) , orthogonal with respect to the weight function \(h(x)\)  on the support interval \(\Lambda \) . These quantities are defined by the corresponding measures of the normalized-to-unity Rakhmanov's probability density \(\rho _n(x) = p_n^2(x)\,h(x)\) . This density function governs the (\(n\rightarrow +\infty \) )-asymptotics of the ratio of two polynomials with consecutive orders [10]}, and describes the quantum-mechanical probability density of the bound stationary states of a great deal of quantum systems in one and many dimensions [1]}, [12]}, [13]}, [14]}, [15]}. Indeed, it happens e.g. that the wavefunctions for the bound states of a large family of non-relativistic quantum-mechanical potentials are controlled by the three canonical HOPs families of Hermite \(H_n(x)\) , Laguerre \(L_n^{(\alpha )}(x)\)  and Jacobi \(P_n^{(\alpha ,\beta )}(x)\)  types [1]}, [5]}. So, the associated Rakhmanov density \(\rho _n(x)\)  may be often interpreted as the charge and/or the matter density of single-particle quantum systems. Consequently, the spreading measures of the HOPs characterize different fundamental and/or experimentally measurable properties of physical and chemical systems.
